I would try a power washer - just test on a hidden area first. It works well on our exterior limestone - which is a soft stone.

illinigirl - the very best thing in the world for drying out a shower is the Costco microfiber towels sold in the auto department. I keep one in my shower and the walls and door are bone dry after a wipe down. It's so much a habit now I don't even think about it. No water spots on the tile or glass or the shower head or handle. As for the fireplace, are you talking about the inside of the firebox? Unless you want it to look like it;s never been used, let it 'age' naturally with the fires. Otherwise it'll look like a too white, too new pair of Keds! Here is a link that might be useful: Costco Microfiber Towels

A fortified thinset mortar will work well. However, on the edge adjoining the firebox opening, depending how you trim it, you should fill any gaps/exposed mortar with high temperature mortar. This is basically a standard mortar mix with fire clay mixed in (which is readily available).

As I understand things, manufactured stone (such as from Eldorado Stone) can come in panels / blocks that comprise multiple simulated stones, and these panels come in predefined lengths. So, how do you deal with needing to cut the stone and expose the cut edge when the stone sits proud of the fireplace itself, such as in this picture or this other picture? Seems like you can't just cut willy nilly because you might interrupt the well-planned lines already in the larger piece. Plus the cut edge itself will be a bit unnatural (too smooth) -- maybe not too noticeable, I dunno.